Arc Permutations

TL;DR

This paper explores the combinatorial structures of arc and unimodal permutations, revealing their pattern avoidance properties, affine group actions, and connections to Young tableaux, leading to new character formulas.

Contribution

It characterizes arc and unimodal permutations via pattern avoidance and establishes their links to affine Weyl group actions and Young tableaux.

Findings

Arc permutations are characterized by pattern avoidance.

A natural affine Weyl group action on arc permutations is identified.

A bijection preserving descent sets relates non-unimodal arc permutations to Young tableaux.

Abstract

Arc permutations and unimodal permutations were introduced in the study of triangulations and characters. This paper studies combinatorial properties and structures on these permutations. First, both sets are characterized by pattern avoidance. It is also shown that arc permutations carry a natural affine Weyl group action, and that the number of geodesics between a distinguished pair of antipodes in the associated Schreier graph, as well as the number of maximal chains in the weak order on unimodal permutations, are both equal to twice the number of standard Young tableaux of shifted staircase shape. Finally, a bijection from non-unimodal arc permutations to Young tableaux of certain shapes, which preserves the descent set, is described and applied to deduce a conjectured character formula of Regev.